Coherent dynamics of strongly interacting electronic spin defects in hexagonal boron nitride

Optically active spin defects in van der Waals materials are promising platforms for modern quantum technologies. Here we investigate the coherent dynamics of strongly interacting ensembles of negatively charged boron-vacancy (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$\end{document}VB−) centers in hexagonal boron nitride (hBN) with varying defect density. By employing advanced dynamical decoupling sequences to selectively isolate different dephasing sources, we observe more than 5-fold improvement in the measured coherence times across all hBN samples. Crucially, we identify that the many-body interaction within the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$\end{document}VB− ensemble plays a substantial role in the coherent dynamics, which is then used to directly estimate the concentration of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$\end{document}VB−. We find that at high ion implantation dosage, only a small portion of the created boron vacancy defects are in the desired negatively charged state. Finally, we investigate the spin response of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$\end{document}VB− to the local charged defects induced electric field signals, and estimate its ground state transverse electric field susceptibility. Our results provide new insights on the spin and charge properties of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{{\rm{V}}}}}}}}}_{{{{{{{{\rm{B}}}}}}}}}^{-}$$\end{document}VB−, which are important for future use of defects in hBN as quantum sensors and simulators.

We characterize the coherent dynamics of V − B ensemble using a home-built confocal laser microscope. A 532 nm laser (Millennia eV High Power CW DPSS Laser) is used for both V − B spin initialization and detection. The laser is shuttered by an acousto-optic modulator (AOM, G&H AOMO 3110-120) in a double-pass configuration to achieve > 10 5 : 1 on/off ratio. An objective lens (Mitutoyo Plan Apo 100x 378-806-3) focuses the laser beam to a diffractionlimited spot with diameter ∼ 0.6 µm and collects the V − B fluorescence. The fluorescence is then separated from the laser beam by a dichroic mirror, and filtered through a long-pass filter before being detected by a single photon counting module (Excelitas SPCM-AQRH-63-FC). The signal is processed by a data acquisition device (National Instruments USB-6343). The objective lens is mounted on a piezo objective scanner (Physik Instrumente PD72Z1x PIFOC), which controls the position of the objective and scans the laser beam vertically. The lateral scanning is performed by an X-Y galvanometer (Thorlabs GVS212).
To isolate an effective two-level system |m s = 0, −1⟩, we position a permanent magnet directly on top of the sample to create an external magnetic field B ∼ 250 G along the c-axis of the hBN lattice. Under this magnetic field, the |m s = ±1⟩ sublevels of the V − B are separated due to the Zeeman effect, and exhibits a splitting 2γ e B, where γ e = 2.8 MHz/G is the gyromagnetic ratio of the V − B electronic spin. A resonant microwave drive with frequency 2.76 GHz is applied to address the transition between |m s = 0⟩ ⇐⇒ |m s = −1⟩ sublevels.
The microwave driving field is generated by mixing the output from a microwave source (Stanford Research SG384) and an arbitrary wave generator (AWG, Chase Scientific Wavepond DAx22000). Specifically, a high-frequency signal at 2.635 GHz from the microwave source is combined with a 0.125 GHz signal from the AWG using a built-in in-phase/quadrature (IQ) modulator, so that the sum frequency at 2.76 GHz is resonant with the |m s = 0⟩ ⇐⇒ |m s = −1⟩ transition. By modulating the amplitude, duration, and phase of the AWG output, we can control the strength, rotation angle, and axis of the microwave pulses. The microwave signal is amplified by a microwave amplifier (Mini-Circuits ZHL-15W-422-S+) and delivered to the hBN sample through a coplanar waveguide. The microwave is shuttered by a switch (Minicircuits ZASWA-2-50DRA+) to prevent any leakage. All equipments are gated through a programmable multi-channel pulse generator (SpinCore PulseBlasterESR-PRO 500) with 2 ns temporal resolution.
We remark that in order to efficiently drive the V − B spin, the strength of the microwave pulse is set to Ω p = 83 MHz in our experiment, corresponding to a π 2 -and π-pulse length as short as 3 ns and 6 ns respectively. The AWG we use has a sampling rate 2 GHz (0.5 ns temporal resolution), sufficiently fast to generate high-fidelity pulses to control the spin state of V − B ensemble. In this section, we derive the dipolar interacting Hamiltonian of the V − B ensemble described by Eq. (1) from the main text. In the laboratory frame, the spin dipole-dipole interaction between two V − B defects can be written as: where J 0 = 52 MHz·nm 3 , r andn denote the distance and direction unit vector between two V − B centers, andŜ 1 and S 2 are the V − B spin-1 operators. Our experiments only focus on an effective two-level system {|m s = 0⟩, |m s = −1⟩}, so the spin operators in the restricted Hilbert space are: Also, we can define the spin raising and lowering operators: and rewrite spin operators in terms of the raising and lowering operators: Then we can expend the dipolar interaction in Supplementary Eq. (1) as: For each V − B center, there is a splitting ∆ = 2.76 GHz between the two levels |m s = 0⟩ and |m s = −1⟩ along the z direction (under a external magnetic field ∼ 250 G). Therefore, the evolution driven by ∆S z is worth to be noted. Consider a quantum state |ϕ⟩ in the rotating frame |φ⟩ = e −i∆S z t |ϕ⟩. If we apply Schrödinger equation: Then we can define dipolar interaction Hamiltonian in the rotating frame: and the spin operators in the rotating frame: In the rotating frame, rewrite the dipolar interaction Hamiltonian (Supplementary Eq. (5)): which can be simplified tõ Since we are interested in spin-spin interaction dynamics with energy scale J 0 /r 3 ≈ 1.8 MHz that is much smaller than the splitting ∆ ≈ 2.76 GHz, we are able to drop the last six time-dependent terms and only keep the energy-conserving terms under the rotating-wave approximation. Additionally, considering n 2 x + n 2 y + n 2 z = 1, we get We can rewrite the interacting Hamiltonian using normal spin-1 2 operators Specifically, we convert the effective two-level spin-1 operators to spin-1 2 operators, S x = √ 2S x , S y = √ 2S y , S z = S z + 1/2, and plug them into Supplementary Eq. (11), where A = 3n 2 z − 1 is the angular dependent factor. To derive the dipolar Hamiltonian of the entire V − B spin ensemble, we simply sum up the interactions between every pair of V − B spins: where A i,j and r i,j represent the angular dependence of the long-range dipolar interaction and the distance between the i th and j th V − B centers.

Supplementary Note 2.2. T1Independence from Dipolar Interaction
We can also re-write the interaction Hamiltonian Supplementary Eq. (14) using raising and lowering operators, S + i and S − i , From this form, we can see that dipolar interaction can lead to spin flip-flop between two nearby V − B (|m s = 0⟩⊗|m s = −1⟩ ⇐⇒ |m s = −1⟩ ⊗ |m s = 0⟩). However, when measuring ensemble T 1 , we characterize dynamics of total spin polarization across the entire V − B ensemble, Σ i ⟨S z i ⟩, which remains unchanged under dipolar flip-flop. Therefore, T 1 is not expected to have a dependence on V − B concentration ρ. Our experimental observation of T 1 decreasing with increasing ion dosages may be attributed to the presence of lattice damage during the implantation process or local charge state hopping [1].

Supplementary Note 3. SWEEPING PULSE INTERVAL VERSUS SWEEPING PULSE NUMBER IN T2 MEASUREMENT
To measure the coherent timescales, T XY8 2 and T D 2 , we choose to fix the time interval between pulses to be τ 0 = 4 ns, much smaller than the correlation time of the noise environment, and increase the number of pulses for each subsequent data point. Importantly, the purpose of this method is two-fold: (1) By fixing the time interval between pulses, the center frequency of the noise filter function of the applied sequence is fixed during the measurement. Consequently, this allows us to sweep the measuring pulse sequence length while avoiding hitting the unwanted resonances due to the hyperfine coupling between V − B and the nearby nuclear spin bath; (2) Given the short-lived coherence of V − B , it is also crucial to obtain enough data points at the early timescale to capture the decoherence decay profiles. By fixing τ 0 to a small value, we can collect more points at the beginning to better characterize the coherent timescales.
For comparison with the fixed pulse interval method in the main text, we have also performed measurements of XY-8 on sample S 3 by fixing the pulse number at N 0 = 8 and N 0 = 16 while increasing the pulse intervals (see Supplementary Fig. 1). We observe that, for both cases, the XY-8 coherent timescale is shorter than the XY-8 measurement with a fixed pulse interval. This is not surprising as we expect the XY-8 sweep τ timescale to approach the XY-8 sweep N timescale at a large enough pulse number N 0 . However, at N 0 = 16, the first data point of the decay profile (corresponding to τ = 2 ns) is already at 128 ns due to the finite duration of the pulses, which is on the same order of the decay timescale one extract from the fitting. Therefore, we choose to fix the pulse interval time and increase the number of pulses for measuring the coherent dynamics of V − B throughout this work. Comparison XY-8 coherent timescales measured on sample S3 with the highest ion implantation dosage using three different methods. The first two are to fix the XY-8 pulse number at N0 = 8 and N0 = 16 while increasing the pulse intervals. The third method is to fix the time interval between pulses to be τ0 = 4 ns while increasing the number of pulses, and this is the measurement technique we choose to use. Here dashed lines are data fitting with single exponential decays. Error bars represent 1 s.d. accounting statistical uncertainties.  π | x π/2 | x π/2 | y π | y τ 0 Fig. 3. Dynamical Decoupling Sequences a) Sequence schematic for laser and microwave. The duration between polarization and detection, τMW, is fixed around 3500 ns to account for the effect of T1 relaxation on the T2 measurement. b) Pulse sequence for spin echo. The rotations along the positive x and y axes are plotted above the line, while the rotations along the negative axes are plotted below the line. π-pulse duration, τπ, is fixed at 6 ns with adjusted microwave power referencing to the Rabi oscillation recorded Fig. 2. Here we sweep the evolution time t to measure the coherent timescale. The final π 2 pulse along the ∓y axis is applied for differential measurement. c) Pulse sequence for XY-8. The sequence repeats itself every 8 pulses, and we take a measurement every 4 pulses to increase the number of data points. The interval between every adjacent pulse, τ0 is fixed at 4 ns. We extract the timescale by increasing the number of sequences, N . d) Pulse sequence for DROID. Here we adopt a truncated version of the original DROID sequence to increase the total number of data points [2]. τ0 is also fixed at 4 ns, but note that there is no interval between the two adjacent π 2 pulses. N is swept to measure the DROID coherent timescales.